Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $x = \dfrac{5n}{6n - 20} \div \dfrac{7n}{2(3n - 10)} $
Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{5n}{6n - 20} \times \dfrac{2(3n - 10)}{7n} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 5n \times 2(3n - 10) } { (6n - 20) \times 7n } $ $ x = \dfrac {5n \times 2(3n - 10)} {7n \times 2(3n - 10)} $ $ x = \dfrac{10n(3n - 10)}{14n(3n - 10)} $ We can cancel the $3n - 10$ so long as $3n - 10 \neq 0$ Therefore $n \neq \dfrac{10}{3}$ $x = \dfrac{10n \cancel{(3n - 10})}{14n \cancel{(3n - 10)}} = \dfrac{10n}{14n} = \dfrac{5}{7} $